An energy-driven approach to linkage unfolding

Authors:
Jason H. Cantarella;Erik D. Demaine;Hayley N. Iben;James F. O'Brien
Affiliations:
University of Georgia;Massachusetts Institute of Technology;University of California, Berkeley;University of California, Berkeley
Venue:
SCG '04 Proceedings of the twentieth annual symposium on Computational geometry
Year:
2004

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Abstract

We present a new algorithm for unfolding planar polygonal linkages without self-intersection based on following the gradient flow of a "repulsive" energy function. This algorithm has several advantages over previous methods. (1) The output motion is represented explicitly and exactly as a piecewise-linear curve in angle space. As a consequence, an exact snapshot of the linkage at any time can be extracted from the output in strongly polynomial time (on a real RAM supporting arithmetic, radicals, and trigonometric functions). (2) Each linear step of the motion can be computed exactly in O(n2) time on a real RAM where n is the number of vertices. (3) We explicitly bound the number of linear steps (and hence the running time) as a polynomial in n and the ratio between the maximum edge length and the initial minimum distance between a vertex and an edge. (4) Our method is practical and easy to implement. We provide a publicly accessible Java applet [1] that implements the algorithm.